Finite Group Databasebeta

← #13167492

Subgroups — #13167492

Order 1728.

590 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88941C212yesmaximal, complemented
2#11723697C211yesmaximal, characteristic, complemented
3C4×#6295C211yesmaximal, characteristic, complemented
4C3×C4⋊#622C211yesmaximal, characteristic, complemented
5#11722793C212yesmaximal, complemented
6#10593417—31nomaximal
7C2×C3×C4⋊(C3⋊D3)—21no—
8C2×C3×C4×C3⋊D3—21no—
9C2×#6295—21no—
10C2×#6295C2212yes—
11C2×#6295C2211yescharacteristic
12C2×C3×#622C2212yes—
13#88941—24no—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15#88815C2211yescharacteristic
16#88814—21no—
17C2×#3774—32no—
18#63914—31no—
19#63911—61no—
20C4×#626—61no—
21C4×#625—31no—
22C4⋊#622D311yescharacteristic, complemented
23#63739—32no—
24C22×C3×C3⋊D3—21no—
25C22×C3×C3⋊D3—22no—
26C2×C33×C4D411yescharacteristic
27C2×C3×C3⋊Dic3—21no—
28C2×C3×C3⋊Dic3D411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C3×C4⋊(C3⋊D3)—42no—
31C3×C4×C3⋊D3—42no—
32#6295—22no—
33#6295—21no—
34#6295—22no—
35#6295—42no—
36#6295D412yescomplemented
37C3×#622—22no—
38#4843—91nomaximal
39C2×C4⋊(C3⋊D3)—21no—
40C2×C4×C3⋊D3—21no—
41C2×C3×D12—61no—
42C2×C3×D12—61no—
43C2×C3×D12—121no—
44C2×C3×C4×D3—61no—
45C2×C3×C4×D3—61no—
46C2×C3×C4×D3—121no—
47C2×#626—61no—
48C2×#626—62no—
49C2×#626—62no—
50C2×#625—31no—
51C2×#625—32no—
52C2×#625—61no—
53C2×#622C2×D312yes—
54#3774—64no—
55C4×C3⋊Dic3C2×D311yescharacteristic
56C3×C4×Dic3—61no—
57C3×C4×Dic3—61no—
58#3724—31no—
59Dic3⋊Dic3—61no—
60#3722—61no—
61(C2×D3)⋊Dic3—61no—
62(C2×D3)⋊Dic3—61no—
63C22×C33C2×D411yescharacteristic
64C2×C3×C3⋊D3—22no—
65C2×C3×C3⋊D3—41no—
66C2×C3×C3⋊D3—41no—
67C2×C3×C3⋊D3—42no—
68C2×C3×C3⋊D3—42no—
69C2×C3×C3⋊D3—42no—
70C33×C4D812yescomplemented
71C3×C3⋊Dic3—21no—
72C3×C3⋊Dic3—41no—
73C3×C3⋊Dic3C2×D411yescharacteristic
74C3×C3⋊Dic3#4111yescharacteristic
75#940—181no—
76C4×#67—181no—
77C2×D24—92no—
78#874—91no—
79C4×D12—91no—
80C3×C4⋊C8—91no—
81#821—92no—
82C22×C3⋊D3—21no—
83C22×C3⋊D3—22no—
84C22×C3×D3—61no—
85C22×C3×D3—61no—
86C22×C3×D3—62no—
87C22×C3×D3—62no—
88C22×C3×D3—121no—
89C22×C3×D3—122no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4—41no—
93C2×C32×C4—41no—
94C2×C32×C4D1211yescharacteristic
95C2×C3⋊Dic3—21no—
96C2×C3⋊Dic3D1211yescharacteristic
97C2×C3⋊Dic3C22×D311yescharacteristic
98C4⋊(C3⋊D3)—42no—
99C4×C3⋊D3—42no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—121no—
107C3×D12—122no—
108C3×D12—124no—
109C3×D12—122no—
110C3×C4×D3—122no—
111C3×C4×D3—124no—
112C3×C4×D3—122no—
113#626—62no—
114#626—62no—
115#626—122no—
116#626—122no—
117#626—124no—
118#625—32no—
119#625—61no—
120#625—62no—
121#625—62no—
122#625—122no—
123#622—22no—
124#459—271noSylow
125C2×C33#12111yescharacteristic
126C2×C33C2×D811yescharacteristic
127C2×C33AGL(1,ℤ/8)11yescharacteristic
128C3×C3⋊D3—42no—
129C3×C3⋊D3—82no—
130C2×C3×D4—181no—
131C22×C3×C4—181no—
132C2×#67—181no—
133C2×#67—182no—
134C2×#67—182no—
135C2×D12—61no—
136C2×D12—61no—
137C2×D12—91no—
138C2×D12—92no—
139C2×D12—121no—
140C2×D12—181no—
141C2×C4×D3—61no—
142C2×C4×D3—61no—
143C2×C4×D3—121no—
144C2×C3×C8—92no—
145C3×C42—91no—
146#217—181no—
147(C2×D3)⋊C4—181no—
148(C2×D3)⋊C4—181no—
149C4⋊Dic3—91no—
150Dic3⋊C4—181no—
151C4×Dic3—61no—
152C4×Dic3—61no—
153D24—184no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—41no—
157C22×C32—41no—
158C22×C32C2×D1211yescharacteristic
159C2×C3⋊D3—22no—
160C2×C3⋊D3—41no—
161C2×C3⋊D3—41no—
162C2×C3⋊D3—42no—
163C2×C3⋊D3—42no—
164C2×C3⋊D3—42no—
165C2×C3×D3—62no—
166C2×C3×D3—62no—
167C2×C3×D3—122no—
168C2×C3×D3—121no—
169C2×C3×D3—121no—
170C2×C3×D3—122no—
171C2×C3×D3—122no—
172C2×C3×D3—121no—
173C2×C3×D3—121no—
174C2×C3×D3—122no—
175C2×C3×D3—122no—
176C2×C3×D3—122no—
177C2×C3×D3—124no—
178C2×C3×D3—122no—
179C2×C3×D3—122no—
180C2×C3×D3—122no—
181C2×C3×D3—242no—
182C2×C3×D3—242no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—42no—
186C32×C4—42no—
187C32×C4D2412yescomplemented
188C3⋊Dic3—21no—
189C3⋊Dic3—41no—
190C3⋊Dic3D3×D411yescharacteristic
191C3⋊Dic3#23911yescharacteristic
192C3×Dic3—61no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—62no—
197C3×Dic3—121no—
198C3×Dic3—121no—
199C3×Dic3—122no—
200C3×Dic3—121no—
201C2×D8—272no—
202#127—271no—
203C4×D4—271no—
204C4⋊C8—91no—
205#102—272no—
206C33#45911yescharacteristic, complemented, Sylow
207C23×C3—181no—
208C23×C3—182no—
209C22×D3—61no—
210C22×D3—61no—
211C22×D3—62no—
212C22×D3—62no—
213C22×D3—121no—
214C22×D3—122no—
215C3×D4—362no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—91no—
221C2×C3×C4—91no—
222C2×C3×C4—181no—
223C2×C3×C4—362no—
224C2×C3×C4GO+(4,𝔽2)11yescharacteristic
225#67—182no—
226#67—182no—
227#67—362no—
228#67—362no—
229#67—364no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—121no—
237D12—92no—
238D12—122no—
239D12—124no—
240D12—122no—
241D12—181no—
242D12—182no—
243D12—182no—
244D12—362no—
245C4×D3—122no—
246C4×D3—124no—
247C4×D3—122no—
248C3×C8—182no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32#89511yescharacteristic
262C2×C32C2×D2411yescharacteristic
263C2×C32#90811yescharacteristic
264C3⋊D3—42no—
265C3⋊D3—82no—
266C3×D3—122no—
267C3×D3—124no—
268C3×D3—122no—
269C3×D3—242no—
270C3×D3—244no—
271C3×D3—242no—
272C2×D4—181no—
273C2×D4—271no—
274C2×D4—272no—
275C2×D4—541no—
276C22×C4—181no—
277D8—544no—
278C2×C8—92no—
279C4⋊C4—271no—
280C22⋊C4—541no—
281C42—91no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—182no—
287C22×C3—361no—
288C22×C3—361no—
289C22×C3—362no—
290C22×C3—362no—
291C22×C3—362no—
292C22×C3C2×GO+(4,𝔽2)11yescharacteristic
293C2×D3—62no—
294C2×D3—62no—
295C2×D3—122no—
296C2×D3—121no—
297C2×D3—121no—
298C2×D3—122no—
299C2×D3—122no—
300C2×D3—121no—
301C2×D3—121no—
302C2×D3—122no—
303C2×D3—122no—
304C2×D3—122no—
305C2×D3—124no—
306C2×D3—122no—
307C2×D3—122no—
308C2×D3—122no—
309C2×D3—242no—
310C2×D3—242no—
311C3×C4—22no—
312C3×C4—22no—
313C3×C4—42no—
314C3×C4—42no—
315C3×C4—91no—
316C3×C4—91no—
317C3×C4—181no—
318C3×C4—361no—
319C3×C4#377412yescomplemented
320Dic3—61no—
321Dic3—61no—
322Dic3—61no—
323Dic3—61no—
324Dic3—62no—
325Dic3—121no—
326Dic3—121no—
327Dic3—122no—
328Dic3—121no—
329C32—21no—
330C32—21no—
331C32—41no—
332C32—41no—
333C32#484311yescharacteristic, complemented
334C23—181no—
335C23—182no—
336D4—272no—
337D4—362no—
338D4—541no—
339D4—542no—
340D4—542no—
341D4—1082no—
342C2×C4—91no—
343C2×C4—91no—
344C2×C4—181no—
345C2×C4—362no—
346C2×C4#157⋊D311yescharacteristic
347C8—182no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—362no—
361C2×C3—722no—
362C2×C3#6422411yescharacteristic
363C2×C3#6423011yescharacteristic
364C2×C3C2×#377411yescharacteristic
365D3—122no—
366D3—124no—
367D3—122no—
368D3—242no—
369D3—244no—
370D3—242no—
371C22—182no—
372C22—361no—
373C22—361no—
374C22—362no—
375C22—362no—
376C22—362no—
377C22C2×#157⋊D311yescharacteristic
378C4—91no—
379C4—91no—
380C4—181no—
381C4—361no—
382C4#8894112yescomplemented
383C3—21no—
384C3—21no—
385C3—41no—
386C3—41no—
387C3#1059341711yescharacteristic, complemented
388C2—362no—
389C2—722no—
390C2#1172434011yescharacteristic
391C2#1172434311yescharacteristic
392C2C2×#8894111yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#459
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#88941C2semidirect
2. #11723697C2semidirect
3. C4×#6295C2semidirect
4. C3×C4⋊#622C2semidirect
5. #11722793C2semidirect
22. C4⋊#622D3semidirect
36. #6295D4semidirect
70. C33×C4D8semidirect
187. C32×C4D24semidirect
206. C33#459semidirect
319. C3×C4#3774semidirect
333. C32#4843semidirect
382. C4#88941semidirect
387. C3#10593417semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#889412
#117236972
C4×#62952
C3×C4⋊#6222
#117227932
#105934173
#48439